mathematical modeling of compaction curve using normal

Journal of Engineering Volume 24 February 2018 Number 2

118

Mathematical Modeling of Compaction Curve Using Normal Distribution Functions

Abdul Kareem Esmat Zainal

Assistant Professor

Engineering college- Baghdad University

Email:[email protected]

ABSTRACT

Compaction curves are widely used in civil engineering especially for road constructions,

embankments, etc. Obtaining the precise amount of Optimum Moisture Content (OMC) that

gives the Maximum Dry Unit weight dmax. is very important, where the desired soil strength can

be achieved in addition to economic aspects.

In this paper, three peak functions were used to obtain the OMC and dmax. through curve fitting

for the values obtained from Standard Proctor Test. Another surface fitting was also used to

model the Ohio’s compaction curves that represent the very large variation of compacted soil

types.

The results showed very good correlation between the values obtained from some published

sample tests and the values obtained from curve fitting for both cases of the single curve and

multiple Ohio’s curves.

The easiness of obtaining OMC and dmax. From the results of curve fitting encourage users to

utilize this procedure, in addition to its accuracy.

Keywords: Compaction Curve, Optimum Moisture Content, Maximum dry density, Procter

Test, Peak functions.

انتمثيم انرياضي نمنحنيات انرص باستخذام دوال انتوزيع انطبيعي

عبذ انكريم عصمث زينم

اسرار يساعذ

خايعح تغذاد –كهح انهذسح

انخلاصة

ذسرعم يحاخ انشص تشكم كثش ف ذخظض انهذسح انذح وخاطح ف عهاخ اشاء انطشق، انرعهاخ انرشاتح، انخ.

وذعرثش عهح اداد يحرىي انشطىتح الايثم ووحذج انىص اندافح انقظىي تشكم دقق ي الاهح تكا يحث ذحم انرشتح

والاعرثاساخ الاقرظادح.

ف هزا انثحث ذى اسرخذاو ثلاثح دوال ي انىع انز ثم انرىصع انطثع نهحظىل عه ذثم ساض تاسرخذاو انحاخ

. كا ذى اسرخذاو ىرج انسطح اناسة Proctor Testاناسثح وذطثقها عه انثااخ انسرحظهح ي ذدشتح تشوكرش

Surface fitting ندىعح يحاخOhio Curves وذثهها عه شكم انر ذثم يذي كثش ي اىاع انرشب انطثعح

.Surfaceيعادنح ساضح ذثم سطح

اظهشخ رائح انرثم انشاض وخىد ذطاتق كثش خذا ت انرائح انسرحظهح ي انعادلاخ انقرشحح تعذ يقاسرها تانرائح

ندع انحالاخ انر ذد دساسرها.انزكىسج ف انظادس

ذعرثش يسأنح سهىنح انحظىل عه رائح دققح ي خلال ذطثق انعادلاخ يشدعح خذا كطشقح اسرعال تذهح ع انطشق

الاخشي )انذوح احاا( اضافح ان دقح انرائح انسرحظهح ي خلال انرطثق.


119

1. INTRODUCTION

Compaction is the application of mechanical energy to a soil to rearrange the particles

and reduce the void ratio. The principal reason for compacting soil is to reduce subsequent

settlement under working loads; Compaction increases the shear strength of the soil and reduces

the voids ratio making it more difficult for water to flow through soil. This is important if the soil

is being used to retain water such as would be required for an earth dam.

The compaction is affected by the water content of the soil, the type of soil being

compacted, and the amount of compactive energy used, Das, 2014.

Soil compaction is widely used in geoengineering and is important for the construction of roads,

dams, landfills, airfields, foundations, hydraulic barriers, and ground improvements. Compaction

is applied to the soil, with the purpose of finding optimum water content to maximize its dry

density, and therefore, to decrease soil’s compressibility, increase its shearing strength, and in

some cases, to reduce its permeability.

A typical compaction curve presents different densification stages when the soil is

compacted with the same apparent energy input but different water contents. The water content

at the peak of the curve is called optimum moisture content (OMC) and represents the water

content in which dry density is maximized for a given compaction energy.

Precise calculations to obtain the OMC are reflected on:

i) The cost of water and its transportation and distribution (e.g. road construction).

ii) The strength of the soil due to obtaining the maximum dry unit weight.

Even small amounts of differences may increase the cost of compaction due to the large

amount of earthwork.

2. LITERATURE REVIEW

Kurucuk, et al., 2007, and Kurucuk, et al., 2008, work was on implementing theoretical

prediction of the compaction curve for sand using unsaturated soil mechanics principles.

It highlights the fact that shape of the compaction curve can be predicted using unsaturated soil

mechanics principles. The main insight gained was that the changes in matric suction are not

important for the evolution of the compaction states, but the influence of matric suction on the

material compressibility with respect to net stress is the governing factor determining the

compaction density. Therefore, it can be reasoned that the inverted parabolic shape of the

compaction curves is a direct function of the variation of the material compressibility with a

degree of saturation.

Altun, 2008, mentioned the importance of obtaining precise values for, in the field,

compaction control is commonly carried out by sand-cone and nuclear gauge tests. Whether

conducted in the field or in the laboratory, these tests are intended to determine optimum water

content and dry unit weight parameters, information required for design specifications.

The parameters of field soil densification obtained by various testing methods performed in the

same region are compared: unit weight, water content, and densification percentage are

measured by nuclear density and sand cone tests. The variations in the outcomes of nuclear

density and sand cone tests, namely unit weight, water content, and densification percent, are

recorded.

Horpibulsuk, et al., 2009, showed compaction curves from 16 coarse and 9 fine-grained

soils, which cover all soil types classified by the Unified Soil Classification System are analyzed

to develop the Modified Ohio’s curves. For all soils, the relationships between water content and

degree of saturation on both the dry and the wet sides of optimum are represented by power

functions. Their compaction curves under standard Proctor energy follow the Ohio’s curves. The


120

optimum degree of saturation, ODS, of coarse-grained soils is lower than that of fine-grained

soils.

Prakash, et al., 2015, studied the water content variations along the compaction curve,

where the degree of saturation of compacted soils also varies. In their experimental work, the

study of the variation of degree saturation along the compaction curve for soils with widely

varying clay mineralogical composition subjected to both Indian Standard light and heavy

compaction efforts. It is observed that the variation of the degree of saturation with molding

water content adopted for both light and heavy compaction tests is linear up to OMC. The degree

of saturation of kaolinitic soils has been observed to be less than that of montmorillonitic soils on

the dry side of optimum; whereas on the wet side of the optimum, the degree of saturation of

kaolinitic soils at any molding water content can be more than that of montmorillonitic soils. In

addition, the degree of saturation of compacted soils at optimum condition has been observed to

be a function of the soil clay mineralogy.

Shrivastava, et al., 2016, mentioned that a need is felt to obtain the required compaction

parameters from the basic soil test which are used for the classification of soil namely

Atterberg’s limits, gradation, specific gravity etc.

Basic soil parameters were collected from literature and Artificial Neural Network (ANN)

techniques have been employed on the data collected, as ANN can better model the relation

between compaction parameters and basic soil properties than statistical modeling.

They demonstrate application of five different ANN algorithms like LM (Levenberg-Marquardt),

GDM (Gradient descent with momentum weight and bias learning function), SCG (Scaled

Conjugate Gradient), and CFB (Circulating Fluidized bed) to predict standard compaction

characteristics of varieties of soils with a large range variation in their basic soil properties.

Multiple variable non-linear regression analysis was also carried out, in which establishment of

an empirical relationship for prediction of compaction characteristics of Modified compaction.

Li, 2013, research was to improve the Compaction Forecasting Expert Database (CFED)

by linking moisture-density-compaction energy relationships with shear strength and stiffness

properties to predict and evaluate the compaction performance of geo-materials.

3. PROPOSED EQUATIONS

The compaction curve used to obtain the optimum moisture content (OMC) and

Maximum dry density (or unit weight) after Proctor method ASTM D698. It is widely used in

experimental tests in laboratories.

The shape of the compaction curve is usually a bell-shaped and the functions that may

describe this kind of shape are peak functions or in a more precise words (Single Peak functions)

through the compaction curve sometimes has multi-peaks and may be described by a multi-peak

functions, but this work is devoted only to the single peak compaction curve.

3.1 First Approach

It is required to describe the compaction curve by a mathematical equation, which may be

useful to obtain the (OMC) and the drymax easily and in a more precise way. Three functions are

presented in this work and verified against some published compaction curves.

Three different compaction curves were taken as examples. Fig. 1, 2, and 3 show these

compaction curves as presented by Das, and Sobhan, 2014, Budhu, 2011, and Fredlund, 2004,

respectively. The compaction curves were digitized and the values of the dry density or the dry

unit weight against water content were obtained as shown in Table 1.

Three functions were proposed as single peak functions that can be implemented as a

mathematical model to describe the compaction curves, which are:


121

1- GaussAmp Function, Eq. (1)

( )

(1)

The OMC and dmax can be easily obtained where:

OMC = xc, dmax = yo + A. These values are the constants that can be obtained from any

Curve fitting computer program, (e.g. MatLab, LabFit, Origin, etc.) to mention but a few.

2- Log Gaussian function, Eq. (2)

[ ( ) ]

(2)

where OMC = eB, dmax = A + D

3- Inverse Poly Function, Eq. (3)

( ) (

) (

) (3)

where OMC = xc, dmax = y0 + A

Through these functions, the compaction curve may be described mathematically and the

(OMC) and drymax. may be obtained more correctly and accurately.

Plate 1 shows the results of the application of Eq. (1) for the three compaction curves with the

result parameters that represents the values of the (OMC) and drymax., in addition to the

correlation factor R2.

The fitted curve for each compaction curve is also shown with the original data also displayed.

The calculated values of (OMC) and drymax. is shown against the values obtained from the

references to compare with.

Plate 2 and 3 show the results of the application of Eq. (2) and Eq. (3) respectively and also

shows the curve fitting parameters for each of the compaction curves already mentioned.

3.2 Second Approach

Ohio’s compaction curves describe a variety of test results for most of the soils (about

10,000 compaction tests on different types of soils), Joslin, 1959, as cited by State of Ohio

Department of Transportation, 2010, the results of these tests are shown in Fig. 4.

Horpibulsuk, et al., 2013, also showed the compaction curves of the fine-grained soils,

and the lateritic soils and crushed rocks. They mentioned that all the fine-grained soils (under

standard Proctor energy) follow the Ohio’s compaction curves, Joslin, 1959. All test data were

collected from the Bureau of Rural Road 6, the Department of Rural Roads, and Thailand. The

results are shown in Fig. 5 for the fine-grained soils only just as an example.

In this approach, a 3-D mathematical surface model is found to describe these curves that

make it easy to determine the values of the wet density Fig. 4 or dry unit weight, Fig. 5 by giving

the desired water content and the required curve (which is described by a number), the numbers

follow a sequence as (A=1, B=2, C=3, … etc. for Fig. 4), and (number 1 for the most upper

curve in Fig. 5, the curve number increases as we go down to lower curves respectively).


122

The curves for each figure were digitized, then three of the most suitable surface

equations were applied that relates the wet density to the curve number and the water content Fig

4 and the dry unit weight to the curve number and the water content Fig. 5. They were found to

be of the form:

1- Poly surface fit of 2nd

degree

(4)

2- Mathematical equation

( )

( ) ( ) (5)

3- Rational2D

(6)

where a0, a1, a2, a3, a4, a5, A, B, C, D, z0, A01, B01, B02, B03, A1, A2, A3, B1, and B2 are

constants

3.2.1 Ohio’s compaction curve

Data of curve in figure 4 were subjected to the three curve fitting equations and the results

were as shown:

1- For the poly surface fit of 2nd

-degree equation, the results were:

(7)

where wet =wet density lb/ft3 ,Cno.=Curve number, and w=water content percent

With correlation coefficient R2=0.9727. As shown in figure 6a.

2- For the mathematical equation, the results were:

( )

( ) ( ) (8)

With correlation coefficient R2=0.956, as shown in figure 6b

3- For the Rational2D equation, the results were:

(9)

With correlation coefficient R2 = 0.98438, as shown in figure 6c.


123

3.2.2 Compaction Curves after, Horpibulsuk, et al., 2013

Data of curve in figure 5 were subjected to the three curve fitting equations and the results

were as shown:

1- For the poly surface fit of 2nd

-degree equation, the results were:

(10)

With correlation coefficient R2=0.977. As shown in figure 7a.

2- For the mathematical equation, the results were:

( )

( ) ( ) (11)

with correlation coefficient R2=0.969, as shown in figure 7b

3- For the Rational2D equation, the results were:

(12)

With correlation coefficient R2 = 0.9912, as shown in figure 6c.

For all the proposed equations, when the derivative of each equation with respect to the

water content (w) is equated to zero ( ⁄ =0) or ( ⁄ =0) gives the equation of a line that

represents the connections between all OMC values in the family of the compaction curves.

4. CONCLUSIONS

Results demonstrated to show that reliable parameters can be obtained by applying for

the non-linear curve fitting programs. This deduction is based on the high correlation factor (R2)

shown against each compaction curve data.

The easiness of determining the values of (OMC) and drymax. from fitting parameters

makes this method faster and more accurate compared to using graph paper for example.

Computer programs that apply non-linear curve fitting are widely used, this paper focuses

and sheds the light on the easiness of obtaining the compaction curve parameters accurately.

In addition, the figures show that even a family of compaction curves can be simulated

through mathematical equations. These equations can describe the fitting surface and the value

of drymax can be obtained by supplying the required curve number and the moisture content.

The three equations proposed for fitting single compaction curve and the three equations

proposed for fitting a surface of a family of curves can be further investigated and any other

mathematical equations can be implemented if more precise values can be obtained.


124

5. REFERENCES

ALTUN, Selim, A. Burak G¨OKTEPE, Alper SEZER, 2008, “Investigation of

Parameters of Compaction Testing”, Turkish J. Eng. Env. Sci., 32 (2008) , 201 – 209.

Budhu, Muni, 2011, “Soil Mechanics and Foundations”, Third Edition, JOHN

WILEY & SONS, INC.

Das, Braja M., Khaled Sobhan, 2014, “Principles of Geotechnical Engineering”,

Eighth Edition, SI, Cengage Learning.

Fredlund, Murray, 2004, “SoilVision Theory Manual”, SoilVision Systems Ltd.

Saskatoon, Saskatchewan, Canada.

Horpibulsuk, Suksun, Apichat Suddeepong, Pokin Chamket, Avirut Chinkulkijniwat,

2013, “Compaction behavior of fine-grained soils, lateritic soils and crushed rocks”,

The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. Soils

and Foundations Volume 53, Issue 1, February 2013, Pages 166–172.

Horpibulsuk, Suksun, Wanchai Katkan, and Anek Naramitkornburee, 2009,

“Modified Ohio’s Curves: A Rapid Estimation of Compaction Curves for Coarse- and

Fine-Grained Soils” Geotechnical Testing Journal, Vol. 32, No. 1.

Joslin, J.C., 1959, “Ohio’s typical water-density curves”, American Society for

testing and materials. Special Technical Publication STP 239, 111–118.

Kurucuk Nurses, Jayantha Kodikara, and Delwyn Fredlund, 2007, “Prediction of

compaction curves”, 10th ANZ Conference on Geomechanics, 2: 115–119.

Kurucuk, N., and J. Kodikara, D.G. Fredlund, 2008, “Theoretical modeling of the

compaction curve”, Unsaturated Soils: Advances in Geo-Engineering – Toll et al.

(eds), Taylor & Francis Group, London, ISBN 978-0-415-47692-8.

Li, Cheng, 2013, “Using the Iowa K test and the gyratory compaction test to improve

the Compaction Forecasting Expert Database”, A thesis submitted to the graduate

faculty In partial fulfillment of the requirements for the degree of Master of Science,

Iowa State University Ames, Iowa.

Prakash, K., H. S. Prasanna, and A. Sridharan, 2015, “Variation of Degree of

Saturation Along The Compaction Curve”, 50th Indian Geotechnical Conference 17th

– 19th December 2015, Pune, Maharashtra, India.

Shrivastava, K., P. K. Jain, Maulana Azad, 2016, “Prediction of Compaction

Parameters using Regression and ANN Tools”, International Journal for Scientific

Research & Development| Vol. 3, Issue 11, 2016 | ISSN (online): 2321-0613.

State of Ohio Department of Transportation, 2010, Supplement 1015 Compaction

Testing of Unbound Materials.


125

Figure 1. Compaction Curve after, Das and

Sobhan, 2014.

Figure 2. Compaction Curve after, Budhu,

2011.

Figure 3. Compaction curve after, Fredlund, 2004.

Figure 4. Ohio’s compaction curves

wet density after, Ohio’s state

department of transportation, 2010.

Figure 5. compaction curves dry unit weight for fine

soils , tests conducted as Ohio’s compaction curves

after, Horpibulsuk, et al., 2009.


126

(a) (a)

(b) (b)

(c) (c)

Figure 6. Ohio Compaction Curves a)

Poly2dohiofinal ,

b) mathematical equation, c) Rational 2D.

Figure 7. Compaction Curves after,

Horpibulsuk, et al., 2009. a) Poly2dohiofinal ,

b) mathematical equation, c) Rational 2D.


127

Table 1. Digitized data for the three curves.

Curve of Figure 1

Dry unit weight kN/m3 Water Content %

14.96 5.67

17.65 7.82

18.52 8.97

19.02 11.09

18.66 11.88

17.01 13.86

Curve of Figure 2

Dry unit weight kN/m3 Water Content %

16.923716 5.28161

18.006425 7.27273

18.451093 8.87632

18.334564 11.2771

17.453734 13.7087

Curve of Figure 3

Dry unit weight kg/m3 Water Content

1523.35 0.08331

1537.191 0.10178

1581.517 0.12102

1651.708 0.14213

1710.299 0.16354

1744.05 0.18625

1704.517 0.2059

1608.118 0.22643

1551.755 0.24143


128

Plate 1. Applying function 1 to the three compaction curves.

Graph Values according to Reference

dmax= 19kN/m3 OMC= 10.4%

Curve 1

Fitting Parameters

yo A xc R2

–75.5 94.42 10.7 0.9965

dmax= 18.92

kN/m3

OMC= 10.7%

Curve 2

Values according to Reference

dmax= 18.5

kN/m3

OMC= 10%

Fitting Parameters

yo A xc R2

-2237.1 2255.6 9.904 0.9992

dmax= 18.5

kN/m3

OMC= 9.9 %

Curve 3


dmax= 1744

kg/m3

OMC= 0.186

Fitting Parameters

yo A xc R2

1514.6 231.5 0.1796 0.9180

dmax= 1746.1

kg/m3

OMC= 0.1796

6 7 8 9 10 11 12 13 1414.5

15

15.5

16

16.5

17

17.5

18

18.5

19

Water Content

Dry

Gaussian Curve Fit

Original Data

Fit Line

5 6 7 8 9 10 11 12 13 1416.8

17

17.2

17.4

17.6

17.8

18

18.2

18.4

18.6

18.8

Water Content

Dry

Gaussian Curve Fit

Original Data

Fit Line

0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.261500

1550

1600

1650

1700

1750

Water Content

Dry

Gaussian Curve Fit

Original Data

Fit Line


129




Curve 1

Fitting Parameters

A B C D R2

5.27 2.335 –0.18 13.78 0.999

dmax= 19.05

kN/m3

OMC= 10.33%

Curve 2


dmax= 18.5

kN/m3

OMC= 10%

Fitting Parameters

A B C D R2

2.033 2.24 -0.2 16.52 0.989

dmax= 18.55

kN/m3

OMC= 9.4

%

Curve 3


dmax= 1744

kg/m3

OMC= 0.186

Fitting Parameters

A B C D R2

218.4 –1.74 –0.073 1528 0.953

dmax= 1746.4

kg/m3

OMC= 0.175

6 7 8 9 10 11 12 13 1414.5

15

15.5

16

16.5

17

17.5

18

18.5

19

19.5

Water Content %

Dry

Gaussian Curve Fit

Original Data

Fit Line

5 6 7 8 9 10 11 12 13 1416.8

17

17.2

17.4

17.6

17.8

18

18.2

18.4

18.6

18.8

Water Content %

Dry

Gaussian Curve Fit

Original Data

Fit Line

0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.261500

1550

1600

1650

1700

1750

Water Content

Dry

Gaussian Curve Fit

Original Data

Fit Line


130




Curve 1

Fitting Parameters

yo A xc R2

11.2924 7.81 10.52 0.9995

dmax= 19.1

kN/m3

OMC= 10.52

%

Curve 2


dmax= 18.5

kN/m3

OMC= 10%

Fitting Parameters

yo A xc R2

14.728 3.78076 9.91758 0.9975

dmax= 18.5

kN/m3

OMC= 9.91758

%

Curve 3


dmax= 1744

kg/m3

OMC= 0.186

Fitting Parameters

yo A xc R2

1519.7 221.41 0.17945 0.99

dmax= 1741.1

kg/m3

OMC= 0.179

5 6 7 8 9 10 11 12 13 14 1514.5

15

15.5

16

16.5

17

17.5

18

18.5

19

19.5

Water Content %

Dry

Gaussian Curve Fit

Original Data

Fit Line

5 6 7 8 9 10 11 12 13 1416.8

17

17.2

17.4

17.6

17.8

18

18.2

18.4

18.6

18.8

Water Content %

Dry

Gaussian Curve Fit

Original Data

Fit Line

0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.261500

1550

1600

1650

1700

1750

Water Content

Dry

Gaussian Curve Fit

Original Data

Fit Line

mathematical modeling of compaction curve using normal

Documents